Differentiability problem

Let be differentiable on with . Select and define for . Prove that is a convergent sequence. Hint: To show is Cauchy, first show that for .

Okay, so it's pretty clear that I have to use the mean value theorem and so there exists such that for . Now it seems that I should just replace with and I will arrive at the hint. However, how can I guarantee that for any , ? Is there something I am missing that guarantees this, or is my actual approach incorrect so far? And even if I arrive at the hint, how would I proceed exactly? Thanks.

Let be differentiable on with . Select and define for . Prove that is a convergent sequence. Hint: To show is Cauchy, first show that for .

Okay, so it's pretty clear that I have to use the mean value theorem and so there exists such that for . Now it seems that I should just replace with and I will arrive at the hint. However, how can I guarantee that for any , ? Is there something I am missing that guarantees this, or is my actual approach incorrect so far? And even if I arrive at the hint, how would I proceed exactly? Thanks.

Let be arbitrary. We may assume WLOG that . For, if we're done and if the opposite inequality is true then in the following method we need only interchange the two values. So, we know that for some we have that

Okay, but I'm afraid of that there exists sequences that repeat indefinitely and therefore do not converge. However after constructing a few examples it seems that if it does repeat then the derivative at at least one point is greater than or equal to one. Is there a way to definitely prove this/is it not really important/does it come up in the proof/whatever? Thanks.

Okay, but I'm afraid of that there exists sequences that repeat indefinitely and therefore do not converge. However after constructing a few examples it seems that if it does repeat then the derivative at at least one point is greater than or equal to one. Is there a way to definitely prove this/is it not really important/does it come up in the proof/whatever? Thanks.